Description of experiments.

Subjects viewed the stimulus depicted inthe figure below, each consisting of half a horizontal grating and half of a vertical grating.

Stimuli

In addition, grating halves were chosen randomly to be either red (HSV color space coordinates: 0.497, 0.4/0.9, 0.7) or green (120.23, 0.4/0.9, 0.7), but commensurate halves always had the same color. The stimulus was displayed on a grey background (0.0, 0.0, 0.2) and two unfilled squares and two horizontal and vertical lines were added on top of it in order to help the subject align the two images. Two different levels of saturation were used for each color, 0.4 and 0.9.

One session was divided into six 3-minute trials. Trials were separated by a 90-second resting period. To allow subjects time to adapt to the stimulus and form a stable percept, the first 30 seconds of each trial were not recorded. The association between color and orientation was kept the same within each session, but was randomized across sessions. For example, the vertical grating was red within a session, whereas the horizontal grating was always green. In contrast, saturation was randomized across trials, as was the position of the split gratings (left or right eye), to avoid any dominant eye effects. The randomization was such that within one session, each saturation level appeared in three trials and each grating positioning occurred in three trials.

Each subject of nine subjects was planned to participate in six total sessions three per orientation/color pairing: four subjects finished 6 total sessions(AJ, MA, ZK, ND), three subjects finished 5 sessions (FG, YW, ML), one subject finished 4 sessions (AB) and the remaing one finished 7 session (ZM). More specifically,

The observers were asked to indicate dominance duration of four different percepts in the figure above by holding down four different buttons on a gamepad. Subjects were instructed to press button 1 if they perceive a single-eye percept (a percept composed of a vertical and horizontal part) with a red left half, button 2 for a stimulus with a red left half, button 3 for an fused, entirely green percept, and button 4 for an fused, entirely red percept. When the subjects perceived anything different from the four percepts, they were instructed not to press any button.

Description of the data

The data is given in a single frame with several columns. The subject ID and trial number are listed first. Each trial has an associated color saturation value (0.4 or 0.9). Eact trial is also characterized by which color is associated with the vertical direction. For instance G_V means that the vertical bars are green. The last two columns list the percepts (Tdata.perc): 0 represents transition between two percepts, 1 represents percept 1, 2 represents percept 2 and so on. and the duration of the percept (Tdata.duration) in microseconds, in the order reported by each subject.

data <- read.csv('dataInOneTable.csv')
head(data)
##   X IDs nTrial Conds Sat Tdata.perc Tdata.duration
## 1 1  AJ      1   G_V 0.9          0             48
## 2 2  AJ      1   G_V 0.9          2         696112
## 3 3  AJ      1   G_V 0.9          0         280000
## 4 4  AJ      1   G_V 0.9          3        3600240
## 5 5  AJ      1   G_V 0.9          0          55952
## 6 6  AJ      1   G_V 0.9          4        3688288

From this we computed the ratio of the total time, \(T_{\rm grouped}\), that observers reported a grouped percept out of the time they reported seeing a coherent percept (Percepts 1, 2, 3 and 4) during each trial. We thus recorded for each trial, \(i,\) of each subject, \[ratio(i) = \frac{T_{\rm grouped}(i)}{T_{\rm grouped}(i) + T_{\rm single}(i)}\] .

data.ratios <- read.csv('fused_over_fusedNSingle_ratios.csv') 
names(data.ratios)<- c("ID","Sat","ratio")
head(data.ratios)
##   ID Sat     ratio
## 1 AJ 0.9 0.6585479
## 2 AJ 0.9 0.5770041
## 3 AJ 0.4 0.3014873
## 4 AJ 0.4 0.5236174
## 5 AJ 0.9 0.3669277
## 6 AJ 0.4 0.2498100

In the following analysis we account for the subject and saturation factors, and pool data over all trials of the same subject. Thus the percept reported by pressing button 1 was always had a red left, and green right half, but the orientation (vertical or horizontal) of each half depended on the trial.

Predominance of grouped and single-eye percepts

Predominance of a percept here means the fraction of time that the percept is perceived. We first check if there is a difference in the predominance of grouped percepts. Our hypothesis was that the predominance increases with saturation as the cue relating the two halves of the stimulus became stronger.

We first present the raw ratios in each condition

The predominance of grouped states tends to increase for most observers. We next compute the probability to show that this increase is real. To do so, we partitioned trials based on the color saturation level used for each trial, grouping across all other conditions. We analyzed changes in predominance using a linear Student-t regression model to account for skewness in the data. We included the condition (low/high color saturation) as a covariate and set the degrees of freedom of the t distribution to 4 to provide robust inference while avoiding computational difficulties often encountered when using a prior for the degrees of freedom (Fonseca et al., 2008). Prior distributions for the regression coeffcients were independent and normal with mean 0, and variance 104. We estimated the mean difference in the fraction of time and its 95% credible interval (CI) and the probability that the difference is greater than 0.

The results are presented in the table below. The table suggests that this is a real effect for five subjects (ZK, AJ, ML, AB, MA). In these subjects, the fraction of time they perceive a grouped states goes up about 10% or higher with an increase in saturation.

Ratios of grouped percepts
ID   adiff.   low CI   high CI   prob.  
ZK 0.367 0.256 0.473 0.999
AJ 0.155 0.075 0.236 0.999
ML 0.110 0.003 0.218 0.977
AB 0.085 -0.012 0.182 0.957
MA 0.040 -0.030 0.110 0.867
ND -0.003 -0.127 0.121 0.481
ZM 0.011 -0.098 0.122 0.578
FG 0.006 -0.024 0.034 0.670
YW -0.019 -0.120 0.076 0.354
adiff. = mean at saturation 0.9 - mean at saturation 0.4

The following figure shows the distributions of means at each saturation level for each subject seperately.

The difference for five out of nine subjects are large while the effect for others are nearly negligible.

The detailed mean values are listed in the table below.

Effect of saturation level on dominance duration
Sat=0.4   Sat=0.9
ID    mean    low CI    high CI    aprob.      ID    mean    low CI    high CI    prob.   
ZK 0.211 0.120 0.312 0.999   ZK 0.571 0.510 0.635 0.012
AJ 0.395 0.331 0.456 0.999   AJ 0.551 0.492 0.604 0.044
ML 0.413 0.336 0.491 0.984   ML 0.524 0.442 0.607 0.276
AB 0.436 0.372 0.501 0.974   AB 0.522 0.436 0.607 0.296
MA 0.581 0.520 0.639 0.006   MA 0.624 0.579 0.667 0.000
ND 0.440 0.356 0.521 0.931   ND 0.435 0.336 0.537 0.902
ZM 0.344 0.249 0.438 0.999   ZM 0.359 0.289 0.426 0.999
FG 0.489 0.468 0.512 0.852   FG 0.494 0.472 0.514 0.713
YW 0.448 0.377 0.521 0.926   YW 0.428 0.349 0.501 0.974
aprob. = probability(mean < 0.5)

Saturation level effects on dominance duration and number of visits

We next examine whether this increase in the predominance of grouped percepts is due to an increase in the mean time of grouped percepts reports, a decrease in the mean time single eye percept reports, or an increased probability of switching to a grouped percept (number of visits to grouped percepts).

Increase in color saturation decreases dominance duration of single-eye percepts

We next compared the mean dominance duration for both single-eye and grouped percepts under the two conditions

The following table shows the posterior probability for the difference in the mean duration of the single eye percept reports and grouped percept reports in the low and high saturation conditions separately. A negative value indicates a decrease. The same subjects that show an increased ratios of observing grouped percepts demonstrate a decreased fraction of reports of single-eye percepts. Comparing with the single-eye dominance duration, grouped percept dominance duration do not tend to change with saturation level. Only two subjects, AB and ZM, showed a probable decrease in duration with an increase change in color saturation.

Effect of saturation level on dominance duration
single-eye perc   grouped perc.
ID    adiff.    low CI    high CI    prob.      ID    adiff.    low CI    high CI    prob.   
ZK -0.232 -0.327 -0.137 0.999   ZK 0.054 -0.025 0.136 0.092
AJ -0.269 -0.372 -0.171 0.999   AJ 0.040 -0.050 0.131 0.191
ML -0.339 -0.456 -0.225 0.999   ML -0.013 -0.091 0.066 0.628
AB -0.222 -0.365 -0.079 0.999   AB -0.182 -0.415 0.045 0.942
MA -0.088 -0.202 0.028 0.933   MA -0.039 -0.110 0.033 0.861
ND -0.015 -0.096 0.067 0.639   ND -0.013 -0.143 0.120 0.584
ZM -0.184 -0.349 -0.018 0.985   ZM -0.481 -0.716 -0.234 0.999
FG 0.007 -0.100 0.115 0.451   FG -0.094 -0.207 0.016 0.950
YW 0.182 0.100 0.265 0.000   YW -0.020 -0.074 0.034 0.764
adiff. = mean at saturation 0.9 - mean at saturation 0.4

The following figure shows the dominance duration of single-eye percepts and grouped percepts at each saturation level for each subject separately.

The following figure shows the single-eye dominance durations at different color saturations.

The following figure shows dominance durations of grouped percepts at different color saturations.

Increasing saturation level increases ratios of the number of visits to grouped percepts.

We next use the same data as above compute ratios of the number of visits to grouped percepts.

\[\frac{N_{\rm grouped}}{N_{\rm grouped} + N_{\rm single_eye}}\]

Based on these ratios, we compute the mean for each subject at the each saturation condition. We use the same analysis as in the case of the predominance of grouped percepts to compute the posterior probability of a change in this ratio.

Ratios of number of visits to grouped percepts
ID   adiff.   low CI   high CI   prob.  
ZK 0.352 0.270 0.432 0.999
AJ 0.072 0.021 0.127 0.997
ML 0.065 -0.006 0.134 0.966
AB 0.067 -0.029 0.164 0.917
MA 0.016 -0.029 0.062 0.765
ND -0.012 -0.107 0.088 0.393
ZM 0.040 -0.041 0.121 0.835
FG 0.007 -0.026 0.039 0.672
YW 0.008 -0.066 0.079 0.593
adiff. = mean at saturation 0.9 - mean at saturation 0.4

The results are shown in the following figure. The analysis is consistent with the result of predominance: the subjects ZK, AJ, ML and AB, and to a lesser degree MA, show a consistent increases in the number of visits to grouped percepts.

In summary, in some subjects an increase in color saturation level leads to an increase in the predominance of grouped percepts. This increase is due to two factors: the decrease in the domininace duration of single-eye percepts and the increase in the fraction of visits to grouped percepts.

Changes in transition probabilities

We next examine whether there is a change in the probability of transitions between and among the grouped and single eye stimuli with an increase in color saturation. To estimate the transition probabilities between percept types, we classified percepts into two states: single-eye, S, corresponding to percepts 1 and 2, and grouped, G, corresponding to percepts 3 and 4. For each trial, we converted the data into two binary data sequences: One sequence contained all transitions from state S with transitions from S to S denoted by 1, and from S to G by 0. The second sequence contained transitions from G, those from G to G denoted by 1, and from G to S by 0. We used all data obtained by each subject in a given condition (low/high color saturation) to estimate the transition probability from S to S, and from G to G by using a Binomial model with a Beta(1,1) prior for the transition probability.

The results of the change of transition on color saturation are listed in the table below.

Effect of saturation level on transition
single-to-single   grouped-to-grouped
ID    adiff.    low CI    high CI    prob.      ID    adiff.    low CI    high CI    prob.   
ZK -0.384 -0.432 -0.336 0.000   ZK 0.207 0.145 0.268 0.999
AJ -0.143 -0.196 -0.091 0.000   AJ 0.101 0.049 0.154 0.999
ML -0.078 -0.131 -0.024 0.002   ML 0.084 0.029 0.138 0.999
AB -0.034 -0.101 0.033 0.156   AB 0.060 -0.018 0.137 0.936
MA -0.058 -0.113 -0.001 0.022   MA 0.001 -0.045 0.048 0.530
ND 0.035 -0.011 0.082 0.930   ND 0.080 0.022 0.136 0.996
ZM -0.119 -0.170 -0.069 0.000   ZM -0.070 -0.136 -0.006 0.016
FG -0.007 -0.072 0.059 0.419   FG 0.010 -0.060 0.079 0.608
YW -0.018 -0.076 0.040 0.273   YW -0.051 -0.108 0.007 0.042
adiff. = mean at saturation 0.9 - mean at saturation 0.4

Analyze saperately.

The results are shown in the table below.

Transition probability
sat=0.4   sat=0.9
      mean    low CI    high CI         mean    low CI    high CI   
ZK asingle 0.740 0.710 0.770   single 0.356 0.319 0.395
ZK grouped 0.348 0.297 0.401   grouped 0.553 0.520 0.586
AJ asingle 0.436 0.399 0.473   single 0.293 0.257 0.330
AJ grouped 0.281 0.244 0.320   grouped 0.382 0.346 0.419
ML asingle 0.388 0.349 0.429   single 0.310 0.273 0.349
ML grouped 0.314 0.275 0.355   grouped 0.398 0.361 0.435
AB asingle 0.575 0.527 0.623   single 0.540 0.494 0.587
AB grouped 0.297 0.242 0.356   grouped 0.359 0.307 0.412
MA asingle 0.368 0.328 0.409   single 0.311 0.273 0.350
MA grouped 0.580 0.547 0.613   grouped 0.581 0.549 0.613
ND asingle 0.602 0.569 0.634   single 0.637 0.604 0.669
ND grouped 0.359 0.318 0.400   grouped 0.437 0.396 0.479
ZM asingle 0.703 0.668 0.737   single 0.583 0.546 0.619
ZM grouped 0.265 0.215 0.319   grouped 0.195 0.155 0.238
FG asingle 0.514 0.468 0.560   single 0.508 0.462 0.555
FG grouped 0.445 0.397 0.494   grouped 0.455 0.407 0.504
YW asingle 0.474 0.434 0.514   single 0.457 0.415 0.497
YW grouped 0.480 0.441 0.519   grouped 0.428 0.388 0.470
a `single’ = ‘single to single’; ‘grouped’ = ‘grouped to grouped’

The same results are also shown in the two figures below. The number of transitions to grouped percepts tends to increase as the increase of color saturation.

As color stararation increases, six out of nine subjects (ZK, AJ, ML, AB, MA and ZM) showed a significant decrease on the transition from single-eye to single-eye percepts and rest of subjects showed very little difference; five (ZK, AJ, ML, AB, ND) showed a significant increase on the transition from grouped to grouped percepts, two subjects had quite significant decrease on the transition. The result implies that transitions to grouped percepts increases as the color saturation level increases.

Full distributions of different quantities per subject

Distribution of observation times

This is the distribution of times for each of the percepts under the four conditions. The four percepts are separated into single-eye and grouped groups. We fit a gamma distribution for each condition using max likelihood and the fitdistr function.

The observations are overall well fit by a gamma distribution, although there is some variability in the goodness of fit across distributions.

To get an idea of how the average duration of the grouped and single eye percepts compare across conditions, we can look at the boxplots comparing the distribution of durations. It would be better to show color saturation levels side-by-side, since we want to compare single to single in either color saturation level, same for grouped.

Frequency of individual percepts

The following are the fraction that each percetp is reported of the individual percepts per subject. Is there any significant pattern here?

The overlay of dominance duration distribution of single-eye percepts.

overlay plot of grouped percepts